How can I create a profile and rotate it to create a mesh using Blender 2.72?
You can use a Screw Modifier. Being in Object Mode select your profile and enable the modifier.


You have now a bunch of extra options. Set the desired axis you want your profile to screw along. Set the Angle to 360 degrees to have the whole form (or 180 if you want to have a half of the form). You may also increase the number of the Steps, so if you apply the modifier later you'll have more geometry. Please see this documentation for more details: 
http://wiki.blender.org/index.php/Doc:2.6/Manual/Modifiers/Generate/Screw


You may notice the strange behaviour of the normals after applying the modifier. Recalculate them using a Tool Shelf button or use the Ctrl+N shortcut, while in Edit Mode.




UPDATE: Placing the origin point to make the Screw Modifier work properly (according to Randy's situation).

I started with placing the inner vertices of the profile perfectly on the one axis.
Then I gave the mesh the same rotation and the origin point the same location like on your printscreens.



I entered the Edit Mode and selected two vertices as pictured below. Then snapped the cursor to selection using shortcut: Shift+S-->Cursor to Selected.
I entered the Object Mode and snapped the origin point to 3D cursor using Tool Shelf button (you may also use the shortcut: Alt+Ctrl+Shift+C-->Origin to 3D cursor).
With the origin point set I enabled the Screw Modifier.